MIAMI  President Bush, pushing for a hard-to-find breakthrough on a broad immigration overhaul, appealed to graduating college students in this diverse city Saturday for help in persuading Congress to produce a bill.

Bush gave the commencement address at Miami Dade College, where more than half the students were raised speaking a language other than English. He gave the Class of 2007 an assignment: Tell their elected representatives in Washington to get going on immigration legislation.

"You see every day the values of hard work, and family, and faith that immigrants bring," the president said. "This experience gives you a special responsibility to make your voices heard."

Bush said the immigration system is deeply broken: Employers are not held accountable enough; borders are not secure enough; businesses need workers willing to do low-paying jobs; and the 12 million people estimated to be in the U.S. illegally cannot all be deported and so must be dealt with "without amnesty and without animosity."

"We must address all elements of this problem together -- or none of them will be solved at all," Bush said.

The president also chose the setting of Miami, a center of Cuban exiles opposed to the communist regime of Fidel Castro, to predict that the "day is nearing" when "the light of liberty will shine" again in Cuba.

"In Havana and other Cuban cities, there are people just like you who are attending school, and dreaming of a better life. Unfortunately, those dreams are stifled by a cruel dictatorship that denies all freedom in the name of a dark and discredited ideology," the president said to loud cheers. "The reign of every tyrant comes to an end."

Castro temporarily handed power to his brother eight months ago because of illness. The 80-year-old revolutionary had ruled the communist island nation for 47 years.

With Castro's condition and exact ailment a state secret, his future role has been the source of much speculation. Cuban officials have given increasingly optimistic reports about his health, and there is a growing expectation that he could soon make his first public appearance since falling ill.

The takeover of Congress by Democrats was supposed to be a boon to Bush's goal of a comprehensive immigration overhaul. He wants to establish a temporary worker program for some illegal immigrants and to create a path to citizenship -- albeit a difficult one -- for many.

After all, it was his fellow Republicans, conservatives who reject the president's approach as too lenient toward lawbreakers, that stymied his plans when they controlled Capitol Hill.

But the Democrats' ascendance in January has not necessarily made easier the search for a bill acceptable to a majority.

The Senate passed a plan last May that tracked closely with Bush's wishes. The proposal died in the House, where tough new border security measures were the priority. A get-tough bill authorizing 700 additional miles of fencing along the U.S.-Mexico border was Congress' only accomplishment on immigration.

Since, the White House has highlighted the effectiveness of stepped-up border enforcement while quietly seeking compromise on broader legislation.

So far, however, the only approach that has grown out of those initial talks would be tougher on illegal immigrants than the Senate bill. Its path to citizenship would require fines, trips back home, long waits and hefty penalties. Some conservatives still call this overly permissive.

The president was hoping to give a lift to those efforts with the commencement address in Miami and also by devoting his weekly radio address on Saturday to the topic.

"Only a bipartisan bill will become law, and we are prepared to work with the president and our Republican colleagues to get the job done and get it done right," said Kennedy, a leader for his party on the issue.

Bush said the talks in Washington are beginning to bear fruit.

"I know convictions run deep on the matter of immigration. Yet I am confident we can have a serious, civil and conclusive debate," he said in his weekly radio address aired in the morning.

Most national polls show people in the U.S. are overwhelmingly supportive of an immigration overhaul that would allow those already in the country illegally to stay, work and earn their way to legal status.

In the late afternoon commencement speech at Miami Dade's Kendall Campus, he emphasized the strength in America's diversity. He spoke before 5,000 people, including 1,500 graduates, many of whom are immigrants or children of immigrants. This point was underscored when the names -- and flags -- of the scores of countries represented were announced to cheers at the beginning of the ceremony.

"Maintaining the promise of America requires that we remain an open and welcoming society," Bush said.

Just outside the school, an anti-war demonstration drew several hundred opposed to the president's Iraq policies.

"I want to do anything I can to bring the war to an end," said one of the protesters, 55-year-old Thomas Kreycik, a mental health counselor. "We are throwing away money, we are throwing away the lives of our troops for no justifiable reason."

Earlier Saturday, over a $25,000-a-person lunch at the bayside home of developer Edward Easton on Key Biscayne, just outside Miami, Bush raised campaign cash for the Republican Party.